Chapter 12: Problem 11
In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{1} \int_{0}^{2}(x+y) d y d x $$
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Mass In Exercises 23 and 24, use spherical coordinates to find the mass of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) with the given density. The density at any point is proportional to the distance between the point and the origin.
In Exercises 55 and \(56,\) use a computer algebra system to evaluate the iterated integral. $$ \int_{0}^{2} \int_{x^{2}}^{2 x}\left(x^{3}+3 y^{2}\right) d y d x $$
In Exercises \(1-10\), evaluate the integral. $$ \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}}\left(x^{2}+y^{2}\right) d x $$
In Exercises \(43-50\), sketch the region \(R\) whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. $$ \int_{0}^{1} \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}} d x d y $$
Use spherical coordinates to show that $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sqrt{x^{2}+y^{2}+z^{2}} e^{-\left(x^{2}+y^{2}+z^{2}\right)} d x d y d z=2 \pi$$
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